1. (Network) Coding in Uncertain Networks
Daniel E. Lucani

2. Sources of Uncertainty
Rate uncertainty
user rate
changes with time,
user, other users
Channel
Delay
packets
are
lost
cannot
react
quickly
Interference
collisions

3. Motivations for Research
Rate uncertainty
How much to talk before stopping to listen?
Capacity improvement through simple, distributed algorithms?
Which transmissions have greatest impact?
Channel
Delay
Uncoded rate without additional cost?
Interference
How to perform efficient rateless transmission?

4. Network Coding A new means of conveying information
Throughput gains, robustness against failures and erasures
Routing/traditional network: data can only be forwarded or replicated
Network coding: data is an algebraic entity
In wireless networks
Broadcast advantage
Packet erasures (losses)
Routing A B
Network Coding A
A+B B 4

5. Network Coding In wireless networks Broadcast advantage
Packet erasures (losses)
p1+ p2 + p3 p2 p3 p1 p1 p2
p1+ p2 + p3 p3 p3 p1 p2
p1+ p2 + p3 5

6. How much to talk before stopping to listen?
On Coding for Delay
Rate uncertainty
How much to talk before stopping to listen?
Channel
Delay
Interference
Joint work with:
Muriel Médard
Milica Stojanovic
David Karger

7. Application: dissemination of information
Time-Division Duplexing or ‘Half-Duplex’ channels
How much should we talk before stopping to listen?
Large latency channels
Satellite and deep space communications
Very long distances (~10,000 km and up)
Underwater acoustic communications
Low propagation speed (~1500 m/s)
Long distances (~km)
Previous work for link/broadcast scenarios
ARQ schemes
FEC schemes: block-based (no feedback) or rateless
Hybrid schemes

8. Cases Link 1-to-all BroadcastRx Tx
Mean completion time: close to full duplex (gold standard)
Mean completion energy: close to optimal
Rx 1
Rx 2 …
Rx N
Tx with M packets
1-to-all Broadcast
Mean completion time: can be better than scheduling schemes with full duplex capabilities
Optimization: Hard, but have good heuristics

9. Cases all-to-all Broadcast
Node 1
M1 packets
Node N
MN packets
all-to-all Broadcast …
Node 3
M3 packets
Node 2
M2 packets
Mean completion time: can be better than scheduling schemes with full duplex capabilities
Optimization: hard, but have algorithm+heuristics

10. Description of Scheme: Link
Goal: reliable transmission of a block of data packets Tx Rx
Erasure Channel
ACK i
Header C1 CM …
n bits
g bits
h bits
Coded Data
Generates N
random linear coded packets
Received k coded packets,
only M-i independent
linear combinations M i
M data
packets .
ACK degrees of freedom
required to decode (dofs req’d):
Not particular data packet .
Degrees of freedom
needed by Rx:
Degrees of freedom
needed to decode: M i M i
Note: Choice of Ni, i determines performance of scheme
[ITA’09, Infocom’09, ICC’09]

11. Description of Scheme: Link
Scheme can be modeled as a Markov chain
States: degrees of freedom (dof) required to decode
Transition time ( ) depends on starting state
We determined moment generating function of completion time [ITA’09]
Optimizing to minimize completion time:
No closed-form expressions
are integer-valued: hard
Exploit recursion, performed off-line

12. Minimizing Completion Time or Energy
TDD constraint
Ni, i chosen to minimize
Mean completion time [ITA’09 Infocom’09]
Mean completion energy [ICC’09]
Full Duplex Network Coding: Tx Rx
ACK
Channel
Time
Energy
Pe = 0.8
1.15dB
Pe = 0.8
1.1dB
e.g: M = 10, round trip time = 250 ms, Rate = 1.5 Mbps

13. Effect of Field Size [GLOBECOM’09] n ind. linear combinations neededTx
Rx 1
Channel / Network
(Packet Losses)
n ind. linear
combinations
needed
Inputs random linear
network coded packets
(M original packets)
j coded packets arrive
We are interested in
Modeled as a Markov chain
with transition
probability matrix
Shown that each receiver requires in average less than M + 2
coded packets to decode regardless of field size (q)

14. Systematic coding: a free lunch?
Can we use smaller field size (simpler operations) and still perform close to large field size?
Ans. Yes + reduction in decoding complexity
Systematic network coding:
First M transmissions are the original packets
All other transmissions: random linear combinations
Decoding complexity down from O(M3) to O(Pe3M3)
Pe = 0.1 → 1000 times less operations in average

15. Inter-Layer Coding for Multicast
Rate uncertainty
Channel
Delay
Capacity improvement through simple, distributed algorithms?
Interference
Joint work with:
Muriel Médard
MinJi Kim
Fang Zhao
Shirley Shi

16. Layered Multicast with Network Coding
Directed acyclic graph, unit capacity links
X1 , X2 , X3
Refinement layers
Base Layer s
Multicast Network r1 r2 r3 X1
X1, X2 , X3
n-layer multicast:
If n = 2, multicast all layers to all but one receiver: achieves min-cut with linear codes [Koetter et al ‘03]
Otherwise: min-cut not achievable with linear codes in general
Previous work, e.g. [Wu et. al, ‘08], [Sundaram et. al ‘05]
“Intra-layer” coding and no “inter-layer” coding
Centralized, LP
Single source
Multicast
Multi-resolution: layers

17. Pushback Algorithm [Infocom’10]
Distributed, random linear network coding
Message passing: two stages
Guarantees decodability of base layer
c(m1)
q(v)
c(m2)
q(v) v
c(m4)
q(u)
c(m3) u

18. Conclusions
Sources of uncertainty are not restricted to channel limitations/constraints, but also include network topologies and interactions between network nodes
On Coding for Delay:
Tailoring feedback and coding can reduce mean delay for successful in-order transmission of packets
Coding is designed to minimize delay
Have looked at queueing analysis
Currently using these intuitions to improve satellite networks
Inter-layer coding multicast:
Simple, distributed algorithm to determine liner network codes
Benefits in coding (when done appropriately)
Extension: use a modified version to determine uncoded rate that can be guaranteed to users

19. Extra Slides

20. Effect of field size [GLOBECOM’09]
Computation: Smaller field size, simpler operations
How much do we loose in performance?
Shown that each receiver requires in average less than M + 2 coded packets to decode regardless of field size (q)
If q large, roughly M
If M is large, not much performance degradation

21. Motivations for Research
Rate uncertainty
How much to talk before stopping to listen?
Capacity improvement through simple, distributed algorithms?
Which transmissions have greatest impact?
Channel
Delay
Interference
How to perform efficient rateless transmission?

22. Future work: Midterm goals
Simple distributed MAC layer data dissemination protocol
Online network coding in large latency half-duplex channels
Improvement of pushback algorithm: can we send uncoded packets while maintaining rate?
Scaling laws for underwater networks  scale frequency

23. Future work: Long term goals
General wireless networks that are challenged by high packet losses and large latency
Special interest in underwater and satellite
Trade-off between cost and rate of sending uncoded packets in network coding
Data dissemination in wireless networks
Theoretical bounds for half-duplex systems
MAC level schemes for fast and efficient dissemination
Voice and video transmission in networks and how to best incorporate network coding and feedback in the network

24. Queuing analysis for large latency half-duplex scheme [ISIT’09]
Other Research Topics
Queuing analysis for large latency half-duplex scheme [ISIT’09]
Underwater acoustic networks
Channel models [Oceans’08, JSAC’08]
Fundamental limits [JSAC’08, Asilomar’08, Submitted to IT Trans.]
Joint work with:
Muriel Médard
Milica Stojanovic

25. Results
Proposed first simple tractable model relating transmission power, band, capacity [Oceans’08, JSAC’08]
Proposed approximate channel models for transmission power [Oceans’08, JSAC’08]
Convex for practical purposes [ISITA’08, JSAC’08]
Lower bound to transmission power for multicast [ISITA’08, JSAC’08]
Based on network coding subgraph selection problem
Use (approximate) channel models as cost function
Capacity scaling [Asilomar’08, Submitted to IT Trans.]
Proposed upper bounds for transport capacity for the underwater networks for different setups

26. Underwater Acoustic Channel
Path loss:
Noise :
Parameters l
Spherical geometry k = 2
A(l,f)N(f) f l
Cylindrical geometry k = 1

27. Underwater Acoustic Channel
Power:
where
Capacity (water filling principle):
where optimum band, constant
fc(l) l
l<<1 km
A(l,f)N(f) f
5 km
10 km
100 km
50 km
A(l,f)N(f) f
K(l,C)
fc(l)
A(l,f)N(f) f
fc(l)
K(l,C’)
B(l,C’)

28. Random Linear Network Coding
Generating a random linear network coded packet (CP)
Operations over finite field of size q = 2g.
e.g. g = 8 bits, q = 256 D A T 1
n bits C1 x
g bits + D A T 2 C2 x
Header C1 C2
n bits
g bits
h bits
Coded Data
Coded Data

29. Throughput
Ni, i chosen to minimize mean completion time for channel conditions
e.g: M = 10, Rate = 10 Mbps
Throughput Metric
 = #bits / E[Time]
Increasing latency, favors
network coding TDD scheme
Better performance than
Go-back-N (GBN) and Selective
Repeat (SR) for TDD

30. Energy Per Bit
n = # of data bits per packet
Fixed bit error probability: n  , packet erasure prob. 
Fixed transmission power
<1.4dB
Optimizing for time or energy: similar performance in energy

31. Sensitivity Errors in estimate of probability of erasure
Over /Underestimate
of Pe : similar results
at low Pe
±2dB
±1dB
±3dB
Underestimate of Pe :
Better performance
at high Pe
<0.2dB
<0.67dB
<2dB
<0.84dB

32. Variance of Completion Time
Variance is not continuous
w.r.t packet erasure prob.
Why?
Change in NM
Change in NM
Change in NM
As packet erasure prob.
increases, number of coded
packets sent Ni increases

33. Broadcast: viable scheme
Number of variables:
Reduce to M variables
Optimization:
# of operations depends on # of states, i.e. ~(M+1)N
Develop heuristics to
reduce computation
Worst Link
Combined Erasures
max dof = 3
3,3
3,2
3,1
3,0
2,3
2,2
2,1
2,0
1,3
1,2
1,1
1,0
0,3
0,2
0,1
0,0

34. Description of scheme: two nodes
Erasure Channel
ACK M2
ACK i1
ACK i1
Generates N( , , )
random linear coded packets M1 i1 M2 i2
Generates N( , , )
random linear coded packets i1
Header C1 CM …
n bits
g bits
h bits
Coded Data M2 1
M1 data
packets .
M2 data
packets . . .
Degrees of freedom
needed by Node 2:
Degrees of freedom
needed to decode:
Degrees of freedom
needed by Node 1:
Degrees of freedom
needed to decode: M1 i1 M2 i2 M2 M1 i1
Note: Choice of N(i,j,t), I,j,t determines performance of scheme

35. Description of the scheme
Modeled as Markov chain
# states N (M1+1)N−1(M2 + 1)N−1...(MN + 1)N−1
N absorbing states
# of variables: large, even for two nodes
Structure:
Round robin assignment of Tx: Periodicity
Node “sees” the system as a broadcast problem [Netcod’09] with round trip time variable, depends on initial state
2,2,0
1,2,0
0,2,0
2,1,0
1,1,0
0,1,0
2,0,0
1,0,0
0,0,0
2,2,1
1,2,1
0,2,1
2,1,1
1,1,1
0,1,1
2,0,1
1,0,1
0,0,1
Absorbing
States

36. Step1: Initialize N(i,j,0) = i, N(i,j,1) = j
Algorithm
Step1: Initialize N(i,j,0) = i, N(i,j,1) = j
Step 2: j’ , fix j’ compute N(i,j’,0) i
Step 3: i’ , fix i’ compute N(i’,j,1) j
Step 4: Converges, then STOP
Node 1
Node 2
Fixed value
N(i,j’,0)
Trt + E[ N(i’,j’,1) P(i,j’,0) -> (i’,j’,1) ] Tp
Dofs needed i’ i
Node 1
Node 2
N(i’,j,1)
Fixed, computed in step 2
Trt + E[ N(i’,j’,0) P(i’,j,1) -> (i’,j’,0) ] Tp
Dofs needed j’ j

37. Number of Iterations of Algorithm
M1 = M2 = M
Small number of iterations needed
before converging to a solution
For Pe < 10-3: algorithm converges to initialization values

38. Comparison Schemes: Two node case
TDD using Round Robin:
Node 1
Channel
Node 2 1 … 2 3 M1 1 3 M1 2 M2 1 … 2 M2
Full Duplex using Round Robin:
Node 1
Channel
Node 2 1 2 3 … M1 1 2 1 3 M1 3 1 M2 2 2 1 M2 … 2 1
Full Duplex using Network Coding:
Node 1
Channel
Node 2

39. Greedy algorithm coding horizon for device i
100%
75%
50%
25%
backward healing
device i
forward dissemination
device j

40. Breaking ties Break tie: node with the most knowledge
M Data Packets, K nodes, transmit to N nodes downstream
Break tie: schedule that benefits nodes with less data
M Data Packets, K nodes, transmit to N nodes downstream

41. Objective: minimize dissemination time, arbitrary wireless networks
Assumptions:
Slotted time
Network modeled as hypergraph
Optimal scheme
Involves choosing the sequence of transmissions
Hard problem
Proposed a greedy algorithm
Chooses hyperarcs that maximize impact at each slot
Impact: Avg. # of nodes that benefit from transmission
Breaks ties in favor of transmissions that disseminate information to nodes with small # degrees of freedom 2
l1{2}
l1{2,3,4}
l1{2,3} 4 1 3

42. Dissemination time gains: losses
Fixed number of nodes
Overhearing reduces gain,
but reduces completion time
Gain increases in presence
of erasures

43. Broadcast Challenges: Optimal scheme: (M+1)N-1 variables to optimizeTx
dofs to decode: M
Rx N
M packets …
dofs to decode: M
Rx 2
Rx 1
dofs to decode: M
Challenges:
Optimal scheme: (M+1)N-1 variables to optimize
Viable scheme: Reduce to M variables
Developed heuristics to determine
[NetCod’09, GLOBECOM’09]

44. Completion Time: Broadcast
Worst Link Heuristic:
Compute Ni’s for link
Combined Erasure:
Compute Ni’s for link
Worst Link Heuristic:
Performs close to
optimal

45. Comparison Schemes: Broadcast
Full Duplex Broadcast using Round Robin: Tx
Rxi 1 … 2 3 M
ACK
i-th
Channel
TDD Broadcast using Round Robin: Tx
i-th
Channel
Rxi 1 2 3 9 M
ACK

46. Completion Time: Broadcast
Outperforms TDD
constrained Round
Robin broadcast scheme
(optimal, no coding)
Pe = 0.8
For high erasure
probability:
Outperforms
full duplex
Round Robin broadcast
Pe > 0.3
better than
RR full duplex

47. Sharing information in TDD: all-to-all broadcast
[Allerton’09]
Node 1
M1 packets
Node N
MN packets …
Node 3
M3 packets
Node 2
M2 packets
Setup:
Broadcast assumption
Round robin assignment of channel
Completion time: Mi original packets of each node i are successfully transmitted to all N-1 nodes and receive ACK

48. Description of the scheme
# of variables: large
Structure:
Round robin assignment of Tx: Periodicity
Node “sees” the system as a broadcast problem with round trip time variable, depends on initial state
Algorithm exploits this structure:
Computes #of coded packets to transmit for broadcast (using heuristics) from each node
Couples the effect of other nodes through the average round trip time
Converges in a small number of iterations

49. At moderate Pe: 1dB away from full duplex
Mean Completion Time
M1 = M2 = M
At high Pe:
better than no coding with a full duplex channel
~3dB more than network coding full duplex
At moderate Pe: 1dB away from full duplex
TDD Scheduling
Full Duplex
Scheduling

50. Underwater Rateless Transmission
Rate uncertainty
How to perform efficient rateless transmission?
Channel
Delay
Interference
Joint work with:
Muriel Médard
Milica Stojanovic

51. Underwater Rateless Transmission
Underwater constraints
Large propagation delay, e.g. 1.5 km  1 s
High packet losses
Transmission band
Very limited, typically from ~0Hz to 100 kHz
Bandwidth decreases as distance increases
System design
MAC model: hard to coordinate or perform power control
ALOHA
Use implicit ACK in network coding
Gain information from state of nodes based on their own coded packet transmissions
Reduce power consumption of purely rateless scheme

52. Numerical Results [WUWNET’07, ISITA’08, JSAC’08]
Node deployed in a square of 1x1 km2
Schemes transmit power to reach receiver with some SNR
Gaussian signaling
~3 dB
[WUWNET’07,
ISITA’08,
JSAC’08]
~11 dB

53. Coding for Data Dissemination
Rate uncertainty
Which transmissions have greatest impact?
Channel
Delay
Interference
Joint work with:
Muriel Médard
Milica Stojanovic
Frank H. P. Fitzek

54. Sharing information in Half-Duplex Channels
[RAWNET’09]
Setup:
Optimal scheme
Involves choosing the sequence of transmissions
Hard problem
Proposed a greedy algorithm
Chooses hyperarcs that maximize impact at each slot
Impact: Avg. # of nodes that benefit from transmission
Breaks ties in favor of transmissions that disseminate information to nodes with small # degrees of
freedom

55. Toy example Progressive Base Station: Slots 3M/2 5M/2 2M M
M Data Packets
Greater Impact (Vanilla):
backward healing
forward dissemination
3M/2
M/2 2M M
M Data Packets

56. Dissemination time gains: no losses
Overhearing
of N = 2
20 nodes
20 packets